Finite Dimensional Evolution Research Articles

Characterization of the absorption radical of an evolution algebra using their associated graph

In this paper, we present a method for finding the absorbing radical of a finite-dimensional evolution algebra. Such a method consists of finding the acyclic vertices of an oriented graph associated with the algebra. The set of generators associated with such vertices turn out to be the generators of the absorption radical. As an application we use the absorption radical to study the decomposability of some degenerate evolution algebras.

On properties of some nilpotent evolution algebras and their enveloping algebras

In this paper, we consider [Formula: see text]-dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra.

Properties of Nilpotent Evolution Algebras with no Maximal Nilindex

As a system of abstract algebra, evolution algebras are commutative and non-associative algebras. There is no deep structure theorem for general non-associative algebras. However, there are deep structure theorem and classification theorem for evolution algebras because it has been introduced concepts of dynamical systems to evolution algebras. Recently, in [25], it has been studied some properties of nilpotent evolution algebra with maximal index (dim E2 = dim E − 1). This paper is devoted to studying nilpotent finite-dimensional evolution algebras E with dim E2 =dim E − 2. We describe Lie algebras related to the evolution of algebras. Moreover, this result allowed us to characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras are found.

On the characterization of the space of derivations in evolution algebras

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph, we prove that the space of derivations is zero. For the remaining families of evolution algebras, we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph’s structure impacts in the characterization of derivations for a given evolution algebra. Therefore, our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results, we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible three-dimensional evolution algebras.

Classification of nilpotent evolution algebras and extensions of their derivations

This article is devoted to n-dimensional nilpotent finite-dimensional evolution algebras with maximal index of nilpotency. We are able to find full invariants of these algebras which allows us to classify them. Furthermore, we explicitly describe its associative enveloping algebras. Moreover, we give an affirmative answer to the question: is it possible to extend a given derivation of the nilpotent evolution algebra to its enveloping algebra.

Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

This paper is devoted to the nilpotent finite-dimensional evolution algebras [Formula: see text] with [Formula: see text]. We describe the Lie algebra of derivations of these algebras. Moreover, in terms of these Lie algebras, we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. Besides, all automorphisms and local automorphisms of these algebras are found.

Stability and nonlinear dynamics of solitary waves generated by subcritical oscillatory instability under the action of feedback control

We consider the influence of global feedback control which acts on an oscillatory system governed by a subcritical Ginzburg-Landau equation. Exact solutions corresponding to solitary-wave solutions are obtained. A generalized variational approach is applied for the simplification of the whole problem and its reduction to a finite-dimensional dynamical model. The finite-dimensional evolution model is used for studying the indirect interaction between solitary waves caused by the global control. The stability analysis is held in the framework of the finite-dimensional model. The boundaries of monotonic and oscillatory instabilities are obtained. The basic types of dynamics provided by the finite-dimensional model are described and compared with the results of a direct numerical simulation of the original equation.